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I have been trying to understand some elementary set theory recently, and am trying to understand how the real number line can be defined using the set of rational numbers. In particular, I am trying to understand Dedekind cuts. I understand the definition of a Dedekind cut, and I have no trouble identifying if a cut is Dedekind or not. But I have troubles understanding how this definition is useful.

In a video I saw by Dr. Peyam, he claimed that $\displaystyle \sqrt[3]{2}$ can be defined the following way:

$\displaystyle \sqrt[3]{2}=\{ r\in\mathbb{Q}:r^{3}<2 \}$

I understand that this set is in fact a Dedekind cut because it has (i) no maximum, (ii) contains all rationals less than $\sqrt[3]{2}$ and (iii) is a real, nonempty subset of $\mathbb{Q}$. But I do not understand how we can define a singular number as a set containg infinitely many numbers.

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    $\begingroup$ You certainly cannot define the single number $\sqrt[3]2$ as a single rational number. Remember that until you have established exactly what the definition of an arbitrary real number is, the only numbers that are available to work with are rational numbers. I don't see a generally useful way to define an arbitrary real number without using infinitely many of these numbers. $\endgroup$
    – David K
    Feb 11 at 16:35
  • $\begingroup$ Just out of curiosity, what is your formal definition of $\Bbb Q$? (This might help address (iii).) $\endgroup$
    – Arthur
    Feb 11 at 16:43
  • $\begingroup$ Other near-duplicates: math.stackexchange.com/q/3554031/96384, math.stackexchange.com/q/752220/96384, math.stackexchange.com/q/2753011/96384 and probably more. $\endgroup$ Feb 11 at 16:43
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    $\begingroup$ Oh yes, I do in fact mean less than $\sqrt[3]{2}$. I miswrote, @TorstenSchoeneberg. Sorry for any confusion. I will update the post right away! $\endgroup$
    – naytte2
    Feb 11 at 17:54
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    $\begingroup$ In descriptive set theory, its normal to think of reals as infinitr sequences of natural numbers. It shouldnt be all that strange, since in non set theoretic lens, we often associate individual Reals with Infinitely long decimal expansions. $\endgroup$ Feb 11 at 18:27

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Sometimes the best way to understand a new definition is by understanding its equivalence with an old definition that you are already familiar with. I will assume that you are familiar with the definition of a real number (say, between 0 and 1) as an unending decimal $a=0.a_1a_2a_3\ldots$ (a technical issue here is that one must also identify 1 with $0.999\ldots$ and all similar cases). Here is an observation that may help: the number $a$ splits (or "cuts") $\mathbb Q$ into two subsets: $A_L$ consisting of all the rationals less than $a$, and $A_R$ consisting of all the rationals greater than $a$ (I will ignore for the moment the issue with $a$ itself in the case when it happens to be rational). Now we have $\mathbb Q$ split into two subsets such that if $x\in A_L$ and $y\in A_R$, then always $x<y$.

Conversely, if one has a splitting of $\mathbb Q$ into a "left" set and a "right" set having the property above, then the splitting uniquely determines a real number.

This is all that is meant when one says that a real number "is" a cut on the rationals. You can actually think of it as the pair $(A_L, A_R)$, but this is merely an issue of set-theoretic formalisation that does not affect the way you actually use the real numbers.

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  • $\begingroup$ Ah, I think I get it now. So when we think of the pair $(A_L, A_R)$, the "gap" between the sets is the real number we are looking for? $\endgroup$
    – naytte2
    Feb 12 at 12:02
  • $\begingroup$ @naytte2, exactly. $\endgroup$ Feb 12 at 13:35
  • $\begingroup$ I see. But how can we be sure that one Dedekind cut corresponds to exactly one real number? How do we know our "gap" does not contain any other irrationals? (In the case of irrational numbers, of course, if the number is rational, it's simply part of one of the sets) $\endgroup$
    – naytte2
    Feb 12 at 13:46
  • $\begingroup$ @naytte2, this follows from the fact that the rationals are dense. If there were two irrationals $\alpha$ and $\beta$ in between $A_L$ and $A_R$, we could find a rational number in between $\alpha$ and $\beta$, contradiction. $\endgroup$ Feb 12 at 15:04
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    $\begingroup$ Well it's a bit of a mannerism of set theorists to insist that a Dedekind cut actually is the real number. You can safely ignore that at your level of study. @naytte2 $\endgroup$ Feb 12 at 15:39

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